Phase‐field based topology optimization with polygonal elements: a finite volume approach for the evolution equation

Uniform grids have been the common choice of domain discretization in the topology optimization literature. Over‐constraining geometrical features of such spatial discretizations can result in mesh‐dependent, sub‐optimal designs. Thus, in the current work, we employ unstructured polygonal meshes constructed using Voronoi tessellations to conduct structural topology optimization. We utilize the phase‐field method, derived from phase transition phenomenon, which makes use of the Allen‐Cahn differential equation and sensitivity analysis to update the evolving structural topology. The solution of the Allen‐Cahn evolution equation is accomplished by means of a centroidal Voronoi tessellation (CVT) based finite volume approach. The unstructured polygonal meshes not only remove mesh bias but also provide greater flexibility in discretizing complicated (e.g. non‐Cartesian) domains. The features of the current approach are demonstrated using various numerical examples for compliance minimization and compliant mechanism problems.